A general issue in evolutionary biology is the relative roles played by adaptation and random genetic drift in generating phenotypic variation. Quantitative genetic tools have a long history in addressing this question (Lande 1976, 1977). More recently, quantitative and molecular tools have been coupled, increasing the scope of relative rates tests to infer the action of drift or selection (*Q*_{ST} vs. *F*_{ST}: Spitz 1993; Merilä & Crnokrak 2001). **G** provides us with a powerful tool to move beyond retrospective analysis, and to address more complex questions about the specific phenotypic effects of different evolutionary process. **G** can identify evolutionary constraints and differences among populations in their potential to evolve and specifically predict the direction and rate of phenotypic divergence (adaptive or neutral) (Lande 1979; Cheverud 1984; Zeng 1988; Arnold 1992; Arnold *et al*. 2001; Phillips *et al*. 2001).

#### Evolutionary potential and absolute genetic constraints

Predicting how a population will respond to selection or whether populations will respond similarly are general aims in evolutionary biology, as well as forming the basis of decision-making in conservation management. Neutral markers and univariate quantitative genetic estimators have been used to summarize the capacity of a population for future phenotypic evolution (e.g. Houle 1992; Hard 1995). When appropriately scaled (e.g. Houle 1992), the size of **G** (the sum of eigenvalues: Σλ_{i}) provides a similar descriptor. Genetic variance can almost be considered ubiquitous (Lynch & Walsh 1998; Barton & Partridge 2000), and estimates of the size of **G** are likewise expected to be greater than zero. However, the size of **G** cannot be used to address such questions as which trait value combinations will evolve most rapidly, or whether some phenotypes are evolutionarily inaccessible. Size of **G** is therefore too crude an estimator of genetic potential to be a useful tool.

Dimensionality of **G** is a potentially informative descriptor of evolutionary potential. **G** has, at most, as many dimensions as traits. In the maximal case, each eigenvector is associated with additive genetic variance and there are no absolute constraints on evolution (Kirkpatrick & Lofsvold 1992; Mezey & Houle 2005). Conversely, if **G** has fewer dimensions than traits, there are phenotypes (trait combinations) without additive genetic variation, and which cannot evolve in the population. Therefore, dimensionality of **G** provides an estimate of how many independent traits **G** summarizes, and what regions of phenotypic space are evolutionarily accessible (Fisher 1930; Dickerson 1955; Kirkpatrick & Lofsvold 1992; Orr 2000; Reeve 2000; Mezey & Houle 2005). Recent methodological papers have outlined two possible approaches for determining the dimensionality of **G**: bootstrapping and factor analytic modelling.

Mezey & Houle (2005) determined dimensionality by estimating bootstrap confidence intervals for each eigenvalue: when eigenvalue confidence intervals exclude zero the associated eigenvector describes a phenotypic space associated with additive genetic variance. Applying this method to wing shape variables in *Drosophila melanogaster*Mezey & Houle (2005) observed statistical support for each of 20 possible dimensions of **G**. This suggests both that different aspects of wing shape have largely independent genetic bases, and that no wing shapes are evolutionarily inaccessible to the population.

Factor analytic modelling is an alternative approach for determining the number of dimensions of **G** (Kirkpatrick & Meyer 2004; Meyer & Kirkpatrick 2005). Meyer & Kirkpatrick (2005) estimated the dimensionality of **G** for eight carcass traits measured on beef cattle. For this **G** there was statistical support for only six dimensions (Meyer & Kirkpatrick 2005). Further work is necessary to fully explore the properties of both bootstrapping and factor analytic modelling, and to then determine whether absolute genetic constraints are common, whether they are more likely for particular types of traits, whether populations vary in their constraints, and what underlies the constraint (lack of mutational input or fixation of alleles).

#### Directional evolutionary constraint or bias

Theoretically, **G** influences the direction of both phenotypic adaptation (Lande 1979; Cheverud 1984; Zeng 1988; Arnold 1992) and drift (Lande 1976, 1979; Arnold *et al*. 2001; Phillips *et al*. 2001). In the decades since Lande (1979) published the multivariate breeder's equation there has been considerable debate over the usefulness of this relationship, which depends on **G** remaining stable over the time frame in question (e.g. Turelli 1988). Schluter (1996) sidestepped this issue, examining not the stability of **G**, but the relationship between **G** and the direction of divergence. Using a novel method to test for an association between **G** and the direction of phenotypic divergence Schluter (1996) demonstrated that **G** was associated with the direction of divergence for at least 4 million years. The direction of phenotypic divergence was defined as the first eigenvector of the divergence variance–covariance matrix, **D** (Box 1). Similarity between **G** and the direction of phenotypic divergence is determined by the angle between major eigenvectors of each matrix (Schluter 1996). Blows & Higgie (2003; Blows *et al*. 2004) extended this approach, comparing the entire phenotypic space described by **G** and by **D**, rather than just their major eigenvectors.

The influence of **G** on the direction of phenotypic divergence is expected to decay over time as the population comes under the influence of an adaptive peak (Lande 1979; Via & Lande 1985; Zeng 1988). Schluter (1996) explicitly looked for and found evidence that the direction of divergence becomes less similar to major eigenvectors of **G** as time since divergence increases. This result contrasts with that of McGuigan *et al*. (2005) who also compared **G** with directions of divergence over different timescales. At the deepest level of divergence (*c.* 2 million years ago), allopatric species of rainbow fish (*Melanotaenia*) diverged in a direction closely associated with the major eigenvector of **G**. However, conspecific populations isolated for < 1 million years did not diverge in a direction predicted by major eigenvectors of **G** (McGuigan *et al*. 2005). Therefore, time since divergence cannot be the only factor determining the association of **G** with directions of phenotypic divergence.

Comparisons of **G** with **D** have not explicitly distinguished divergence due to different evolutionary processes, such as selection vs. drift. In laboratory experiments where selection is artificially applied, it is straightforward to statistically distinguish variance attributable to drift from that due to selection. Multivariate analysis of variance (manova) can be used to extract variance components specific to each source of experimental variance. For example, **D** can be estimated for the selection treatment (**D**_{selection}) and for each replicate (population or line) nested within treatment (**D**_{drift}) (McGuigan *et al*. 2005). Nonetheless, **D** has typically been estimated as the variance–covariance matrix of means from each population (e.g. Blows & Higgie 2003).

With natural populations, there are many different aspects of environment and population structure that could be driving phenotypic evolution. The breeder's equation specifically deals with selection, and investigators have generally examined the relationship between **G** and **D** for functionally important traits known or suspected to be under selection. Langerhans & DeWitt (2004) proposed a nested manova approach to investigate repeated evolution. Using nested manova, phenotypic variance due to selection imposed by dominant aspects of the environment is partitioned from variance due to other processes, such as drift and subtle differences in the position of local selective optima.

McGuigan *et al*. (2005) took a similar approach in their study of rainbow fish, using manova to partition phenotypic divergence between water flow habitat, species, and population. Rainbow fish adapt to still vs. flowing water, evolving in body shape, sustained swimming performance and muscle morphology (McGuigan *et al*. 2003). Colonization of lakes by stream fish has independently occurred several times in different species. By comparing **G** to the **D** estimated for each of the different sources of variance McGuigan *et al*. (2005) determined that adaptation to water flow was less strongly associated with **G** than was divergence among species or among replicate populations. This result indicates the directional constraint imposed by **G** depends on the evolutionary process driving divergence (McGuigan *et al*. 2005). McGuigan *et al*. (2005) hypothesized that conspecific populations experiencing the same water velocity diverged primarily through drift, and reported that **D** estimated at this level (populations nested within species and habitat) was approximately proportional to **G**, consistent with predictions for drift-driven divergence (Lande 1979; Arnold *et al*. 2001). These results suggest drift might be more tightly constrained by **G** than is adaptation (McGuigan *et al*. 2005). There is substantial scope to test these hypotheses, particularly in systems in which drift and selection can be partitioned (Langerhans & DeWitt 2004).

Whether or not **G** persistently influences the direction of divergence will depend on the characteristics of the adaptive landscape (Box 1; Fig. 2) (Cheverud 1984; Lande 1984; Brodie 1992; Jones *et al*. 2003, 2004). If the eigenstructure of **G** is similar to that of the adaptive landscape, then both selection and genetic covariation will drive evolution in the same direction, and **G** will be persistently associated with **D** (Arnold *et al*. 2001; Arnold 2003). That is, selection along major eigenvectors of **G** will result in divergence along major eigenvectors of **G** (Fig. 2B). Conversely, selection orthogonal to major eigenvectors of **G** will cause the association between **G** and **D** to decay with time as the population attains the adaptive peak (Fig. 2A).

Blows *et al*. (2004) estimated both **G** and the fitness surface for male cuticular hydrocarbons (contact pheromones) in *Drosophila serrata*. These hydrocarbons form the basis of the mate choice system in the species, and are under strong directional selection through female mate choice. **G** was not aligned with either γ or ω, matrices describing the curvature of the adaptive landscape (Box 1) (Blows *et al*. 2004). Orthogonality of **G** to divergence (e.g. Merilä & Björklund 1999; Badyaev & Hill 2000; McGuigan *et al*. 2005), and/or the fitness surface (Blows *et al*. 2004) could indicate erosion of genetic variance during adaptation, or reveal a genuine lack of congruence between **G** and the adaptive landscape. Identification of common relationships awaits further empirical investigation.

Finally, variation among taxa and/or traits in the evolutionary stability of **G** might contribute to variability in the association of **D** with **G**. Typically, **G** is estimated from a single extant population in a single environment (but see e.g. Begin & Roff 2003, 2004; Blows & Higgie 2003). Point estimates of **G** are implicitly assumed to represent the variation available during divergence. If this assumption is invalid there is little reason to expect **G** to be similar to **D**. The average **G** has been shown to be a better predictor of phenotypic evolution than single estimates of **G**, highlighting the importance of short-term stochastic fluctuations, as well as longer-term evolution (Jones *et al*. 2004). Two issues are involved here: direct environment effects and evolution of **G**.